Question

which of the following is true for f(x) = -2sin(x) - 3?

the range of the function is the set of real numbers -2 ≤ y ≤ 2.

the graph of the function is the graph of f(x) = -2sin(x) shifted 3 units up.

the amplitude of the function is 2.

the period of the function is 4π.

Explanation:

Step1: Recall sine function properties

The general form of a sine function is \( f(x) = A\sin(Bx - C) + D \), where:

• Amplitude is \( |A| \)
• Period is \( \frac{2\pi}{|B|} \)
• Phase shift is \( \frac{C}{B} \)
• Vertical shift is \( D \)
• Range is \( [D - |A|, D + |A|] \)

For \( f(x) = -2\sin(x) - 3 \), we have \( A = -2 \), \( B = 1 \), \( C = 0 \), \( D = -3 \).

Step2: Analyze each option

• Option 1 (Range):

Using the range formula \( [D - |A|, D + |A|] \), substitute \( D = -3 \) and \( |A| = 2 \):
\( [-3 - 2, -3 + 2] = [-5, -1] \). So the range is \( -5 \leq y \leq -1 \), not \( -2 \leq y \leq 2 \). Incorrect.

• Option 2 (Vertical Shift):

The vertical shift of \( f(x) = A\sin(Bx) + D \) is \( D \). Here, \( D = -3 \), so the graph of \( f(x) = -2\sin(x) \) is shifted down 3 units (since \( D = -3 \)), not up. Incorrect.

• Option 3 (Amplitude):

Amplitude is \( |A| \). For \( A = -2 \), \( |A| = 2 \). Correct.

• Option 4 (Period):

Period is \( \frac{2\pi}{|B|} \). For \( B = 1 \), \( \frac{2\pi}{1} = 2\pi \), not \( 4\pi \). Incorrect.

Answer:

The amplitude of the function is 2. (The option stating "The amplitude of the function is 2.")